For every integer $n\ge 2$ let $$P(n)=\prod (\pm \sqrt{1} \pm \sqrt{2} \cdots \pm \sqrt{n})$$ where the product is over all possible permutations of the signs.
- Prove $P(n)\in \mathbb{Z}\;\forall n$
- Let $p_n$ be the largest prime divisor of $P(n)$. Show for each $\epsilon >0$ there is $n_0$ such that for $\forall n>n_0$ we have $$\log_2 \log_2 p_n<\epsilon n$$
This problem is from the magazine Komal, A578, I found it interesting but couldn't do it. Thanks for any help.
I can only offer the solution for the first part, using the identity $(x-\sqrt{a})(x+\sqrt{a}) = x^2 - a$ repeatedly.
In the second part, I do not know how to show $p_n < 2^{2^{\epsilon n}} $ for all $\epsilon > 0$ and $n > n_0$.